Introduction

Once in a while a topic is presented and its usefulness is instantly apparent. You immediately ask yourself how you got through life thus far without knowing
about such an amazing concept.

I haven’t studied math since high school, but I still remember one fateful morning in geometry class that changed my life.

Background

As a youth back in school it wasn’t uncommon when presented with a new topic, to hear comments like “When will I ever use this in life?” I remember having one
of these moments in grade 10 math class, when we spent a solid five months studying parabolas and quadratic equations. In hindsight, the problem was that we weren’t given any context
for how these curves could be useful. The teacher taught that which the curriculum dictated, but any significance of these equations was lost in translation.

The Triangle

I think it was in grade 7 that we were presented with the Pythagorean theorem. With the knowledge of the lengths
of two sides of a right-angle triangle, the theorem explains how to determine the length of the third side.

How useful! With this knowledge you can safely position an extension ladder to clean out the eaves troughs, calculate the shortest distance between two points
on a map, or buy enough wallpaper to cover an angled wall knowing only the dimensions of the other walls in the room.

Note: Americans call these “right triangles”. Please forgive them.

It felt like I was missing out on the full potential of this handy little triangle though. I wasn’t sure what else it could do, but something was missing.

One morning in grade 13 geometry class, the missing pieces presented themselves. The angles! Of course! Sure, the Pythagorean Theorem explained the relationship
between the sides, but we had forgotten about the angles. In grade 7 we accepted that one of the angles is 90° and that all three angles total 180° but we left it at that.

Trigonometry

That fateful trigonometry lesson in grade 13 explained a lot: how to determine any angle in the triangle, knowing the length of two sides; how to determine the length
of any side, knowing the length of one side and one angle; and just for what purpose those mysterious tan, cos, and sin buttons on my calculator were.

Let’s review.

All triangles have three sides.

In a right-angle triangle, two of those sides meet at a right angle (90°).

The third side, called the hypotenuse, completes the shape.

Here’s a triangle. The little square opposite the hypotenuse denotes that it is a right-angle triangle (if it’s not present, we can not assume the angle is 90°).

To make the rest of this lesson easy to understand, let’s draw a little guy next to one of the points (called vertices)
of the triangle. His name is Dieter. The usefulness of this simple act cannot be overstated, because now we can use relational terms to describe the two unnamed sides. The side
next to Dieter, the one adjacent to him is called… the adjacent side. The side opposite him is… you guessed it, the opposite side.

Now that the sides are named we can apply some handy formulae. Don’t worry; they’re as simple as Pythagoras’ famous formula. We’ll use the symbol θ (Greek
letter theta) to represent the angle closest to Dieter. We’ll abbreviate opposite and adjacent to OPP and ADJ.

The angle of incline is about 20.4°. The inverse of the sine function is called the arcsine and is usually denoted by the negative exponent (don’t let it confuse you).

Circles

If you think about it, every point on the circumference of a circle can be expressed in terms of triangles.

Here’s a task I’ve had to do many times, like when I wrote the Shoe Label
preview for Mabel’s Labels. When placing the letter d in the illustration, what are the x and y coordinates if we need to rotate
the text 30° and the radius is 50?

Demonstration

The demonstration application uses Silverlight to demonstrate the concepts of right-angle triangles and tangential functions to animate
an elastic collision between a moving ball and a fixed post.

There isn’t anything particularly interesting about the application, but you can see these concepts expressed in code. The animation doesn’t take advantage
of Silverlight’s animation features, so it may be less than smooth.

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Conclusion

If you remember three simple formulae and learn to use three buttons on your calculator, you’ll be able to amaze the other 72%¹ of the population who can not.

Addendum

¹ When I originally wrote this article, I made up a statistic for the percentage of people that are able to use these formulae.
I became curious about the actual number, so I initiated a research project.

In my experiment I paid 547 people from around the world (using Amazon Mechanical Turk)
USD 0.01 to solve Dieter’s problem above. The answers were multiple choice, including six numeric answers in addition to “I have no idea how to solve this.”
and “I remember doing this in school, but I forget how to solve it.”. A penny isn’t much money, but they weren’t told
that they would get paid whether or not the solution was correct. They were given five minutes to solve the problem. For those
with a working knowledge, the problem could easily be solved in about thirty seconds. Some might have gone to Wikipedia
and figured it out; others might have just clicked anything to move on to the next HIT. The results were as follows:

8% “I have no idea”

40% “I remember doing this in school”

24% An incorrect answer or guess

28% The correct answer

Acknowledgements

These things take a long time to write. If you like it, please rate it.

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About the Author

Yvan Rodrigues has 25 years of experience in information systems and software development for the manufacturing sector. He runs Red Cell Innovation Inc./L'innovation de Globules Rouges, a consulting company focusing on efficiency and automation of manufacturing and business processes for small businesses, healthcare, and the public sector. He is a Certified Technician (C.Tech.), a professional designation granted by the Institute of Engineering Technology of Ontario (IETO).

Thanks for this article. I feel that math skills and "Number Sense" are being neglected more and more in schools, and are beginning to look like a dying art.

I learned Trig in college, and as a draftsman and programmer, I use them nearly every day, yet I'm saddened when I see new draftsmen who can't do trig and don't want to learn. They can just draw a triangle in a CAD program and measure it.

These are skills that should not be left behind, replaced by software. Math is a fascinating journey, and I thank you for stopping to appreciate one of the steps on that journey.

I think one of the things that this article brings to attention is that almost anything can be taught (and therefore learned) more effectively when well thought out. There were lots of things in school, even math, that I never was able to understand, and more recently I have look up in Wikipedia or another learning resource and have realized that it was the way it was taught that was the problem, not the subject matter.

While I did really well with trig in grade 13, I did relatively poorly with calculus. I have since gone to other resources and have found wonderfully simple explanations.

I am very jealous of kids today who have these amazing resources available to them.

I have to agree that the teaching methods can make all the difference, especially with the "boring" subjects. I've always wanted to put together a "road show" to go to schools and put on science and math demonstrations, hoping to spark an interest in kids. Maybe after I retire . . .

I too had a similar problem with calculus. The professor didn't teach - he would answer questions, if he deemed them worthy. I was doing okay until we hit derivatives, then it went from math to magic in one chapter, and I couldn't seem to formulate any questions that he considered worth his time. We started the class with about 40 people, and at the end, there were around 10, and most of them had already failed it once. That's been about 30 years ago, now, and thinking about it still makes me angry. I'm finally getting back into it, and would welcome any resources you might have.

I was most impressed by your statistics result on how many don't know how to solve such math problems. I think modern gadgets such as iPods, iPhones and game machines may have contributed to the decrease in interest in math and science among young people. They take away people's time and make math look boring. Many of them don't have the brain power of some ancient people, such as Archimedes.

I know people who can recite the value of π to 20 or 30 digits, but have no idea what it means i.e. the ratio of a circles circumference to its diameter. In fact, I don't know that they ever taught us that in school explicitly. It was just a magic number that fit into all of the equations.

Excellent article. I'm an engineer and really into mathematics, so please allow me to add a bit more information, some of which you alluded to in your article.

You wrote:

sin θ = OPP / HYP
cos θ = ADJ / HYP
tan θ = OPP / ADJ

That is completely correct, however, that is not the formal mathematical definition for the cosine and the sine.

The formal mathematical definitions of the cosine and the sine is related to the unit circle. You show in your circle diagram how all right triangles with a fixed length hypotenuse of r can by seen on a circle with radius r.

The unit circle in the x, y Cartesian coordinate system is defined by the equation:

x^2 + y^2 = 1

Where x^2 means "x squared".
It's easy to see the Pythagorean theorem in the formula.

At each point on the circle, there is an angle formed between the point on the circle and the x axis. The x value of the point on the circle is the cosine of that angle and the y value of the point is the sine of the angle. In other words, all cosine and sine values are points on the unit circle. Look at the triangle in your circle diagram. The angle that I am referring to is the angle in the triangle that is closest to the center of the circle.

The formal definition of the tangent of the angle P is:

sin P
tan P = --------
cos P

And, of course, tan P is undefined when the cosine of P is 0, which happens at angles that are an odd multiple of plus or minus 90 degrees. The tangent approaches infinity as it approaches these angles.

That all leads to the triangle formulas for the cosine and sine you showed. More can be derived from the circle formula that is not directly related to triangles.

By the way, a circle with radius r with the center at the origin (the origin being the point (0, 0)), is defined by the equation:

x^2 + y^2 = r^2

Each point on that circle has (x, y) values of (r cos P and r sin P)

Finally, as a point of interest, a circle with center (h, k) and a radius of r is defined by:

(x - h)^2 + (y - k)^2 = r^2

So, when x is h, x - h is 0, and when y is k, y - k is 0. So subtracting h from x and k from y respectively shifts the origin of the simple circle formula from the point (0, 0) to the point (h, k).

----------------

While angles in trigonometry are sometimes measured in "degrees", in mathematics, angles are often measured in "radians."

Radian measure uses the distance going counterclockwise around the circumference of the unit circle and starting on the x axis at the value x = 1.0. Since the circle has a radius of 1, the distance around the circle is 2 Pi (where Pi is approximately 3.1415926535 ...).

You can add or subtract any multiple of 2 Pi from any angle in radians and the angle stays the same, just as you could add or subtract any multiple of 360 from the angle in degrees and the angle would stay the same.

I've never heard trig described in terms of circles, (until now) but it makes a lot of sense, and also explains the idea of radians (always wondered about that one). Thanks.

I work for a steel fabricator, and I love creating mathematical models to use in programs that make our work more efficient. I have a head-scratcher that deals with the intersection of a circle and an ellipse that I'd love to run by you, if you'd consider looking at it. I took it to my local college, and received nothing but confused looks from people who should know a lot more than me.

If you still need help with your problem, send me e-mail at:
xyzBillxyz_xyzHallahanxyz@xyzmailxyz.xyzcomxyz
Just remove all the xyz strings from that e-mail address. (The xyz strings were just added to hopefully defeat automatic e-mail collectors used by spammers).

We just had this at school today, but I didn't understand much. I have been coding half my life so it's like a second language to me. I really wish there was a book called "Math explained in code" or something, because some reason math make much more sense in code than it does on the blackboard.